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Alright, here's a physics problem for you.

If there was a treadmill of infinite length, and the width of a runway, and it was designed to MATCH the speed of the wheels of an airplane, would the airplane be able to take off the ground?

Assume the wheels are fully contacted with the ground until the moment of takeoff (completely off the ground).

This was being discussed on another message board, but it will probably be easier for all YOU smart folk.
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Yeah, it will.

Seems wheels-on-treadmills is the same low-friction environment as, say, a waterplane will get, or a plane on skis would experience.
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Of course, the wheels don't make the plane fly, the wings do...
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Airplanes are lifted due to air currents. The next time you're in a car, put your hand out the window and angle it upward. Your hand gets raised. The opposite is true when you angle your hand downwards. If said airplane was in a vacuum, it wouldn't lift. If it were in somewhere on Earth, it would. QED.
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Yeah, that's what I mean by the wings. The airplanes use the air currents with its wings, not the wheels.
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Twasbrillig wrote:Airplanes are lifted due to air currents. The next time you're in a car, put your hand out the window and angle it upward. Your hand gets raised. The opposite is true when you angle your hand downwards. If said airplane was in a vacuum, it wouldn't lift. If it were in somewhere on Earth, it would. QED.

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Airplanes are lifted by the speed of the air relative to the wings. If the plane can't move forward when its on the ground, or water or treadmill, it can't get any lift and won't fly. The treadmill will make it impossible for the plane to get any air speed over the wings ergo it won't fly, unless there is a very strong head wind and it lifts due to that.
Last edited by jestingrabbit on Sat Feb 24, 2007 1:14 am UTC, edited 1 time in total.

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Twasbrillig wrote:Airplanes are lifted due to air currents. The next time you're in a car, put your hand out the window and angle it upward. Your hand gets raised. The opposite is true when you angle your hand downwards. If said airplane was in a vacuum, it wouldn't lift. If it were in somewhere on Earth, it would. QED.

I think the problem is suggesting that, since the treadmill "matches the speed of the wheels", the aeroplane remains stationary with respect to an observer standing beside the treadmill. (It's just 'treading water' as it were). In this case, there would be no airflow past the wings, and no lift.

Of course, this can never happen (not even in theory), since the treadmill does not exert a backward force on the aeroplane*, whereas there is a big kickass forward force exerted by the engines. So the aeroplane must accelerate.

*I think...or maybe it does exert a frictional force. My physics is a bit rusty. In any case it doesn't really matter.

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i think you chose the wrong forum for this problem
i can sorta see why some people think it won't take off, but it doesn't take long to convince them otherwise
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It's aÃ¶ready been discussed, the plane gets its power from the engines not the wheels, the wheels are merely a surface supposed to limit friction, replace wheels with some sort of hover mechanism and the problem is barely changed yet alot easier to see what will occur.

so the awnser is yes the plane will take off
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jestingrabbit wrote:Airplanes are lifted by the speed of the air relative to the wings. If the plane can't move forward when its on the ground, or water or treadmill, it can't get any lift and won't fly. The treadmill will make it impossible for the plane to get any air speed over the wings ergo it won't fly, unless there is a very strong head wind and it lifts due to that.

So that's what I originally thought, too, but the difficulty is what does it mean for the treadmill to match the speed of the wheels? Suppose the plane fires up its engines enough so that if it were on fixed ground, it would be going 100mph. What does the treadmill do?

If it goes backward at 100mph, then the plane still moves forward at roughly 100mph, because unlike a car or train, a plane's thrust comes from pushing on air, not the ground. So the wheels would be spinning as fast as they do when the plane is going 200mph over fixed ground. The wheels are actually moving at 100mph, and the treadmill is going 100mph, even though they're spinning twice as fast as they normally would at that speed. Is that what's meant by matching speed? If so, the plane takes off.

Alternately, what I first imagined is that when the plane fires its engine as if to go at 100mph, the treadmill spins backward at some speed Nmph, such that the plane and wheels are not moving with respect to the fixed ground, but the wheels are spinning forward as if the plane were moving on fixed ground at speed Nmph. I now think this is ridiculous, because of how huge N must be. It would need to apply enough rolling friction to the wheels in order to slow the whole plane down. In real life, I think it would melt the tires first.

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How the hell do you get that typo???

Anyway, yeah, it'll fly. yay
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phlip wrote:http://forums.xkcd.com/viewtopic.php?t=361

Lotsa discussion there.

Sorry, to tell you the truth, I didn't really bother doing a search.

I didn't really expect it to have been posted already!
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Heh.. I think you guys are all looking at this the wrong way around..

The answer to the question as posed is "no, the plane could not take off". Here's my reasoning:

The question states that the treadmill "is designed to match the speed of the wheels". Since there's not enough information given to make angular velocity a useful way to interpret this, the only way I can interpret this is linear velocity. Since talking about the relative linear velocities at the point of contact (assuming no slippage) is pointless (by definition they'll always be the same), there would be no reason to mention this. The only other sensible interpretation I can make of this statement is that points on the outside of the wheels and points on the treadmill will have the same net linear velocity (over their entire respective circuits).

If the plane is moving forward relative to the treadmill, then that forward velocity would also be added to the linear velocity of any points on its wheels, and thus the net wheel-surface velocities would by definition not be the same as the treadmill-surface velocity.

Therefore, saying that the treadmill was designed to match the speed of the wheels is logically equivalent to saying "the treadmill was designed such that the plane cannot move relative to the treadmill" (presumably there's some physical restraint structure designed into the treadmill that just wasn't mentioned?), because any design that did allow it to move would invalidate the speed constraint.

Therefore the plane cannot fly, because it cannot move forward to get the air currents required to lift it. The reason it can't fly isn't because of the treadmill's actual speed (which is really irrelevant), it's because the treadmill apparatus was apparently also designed with some add-on equipment to hold the plane in place. The question tells us so, just not in so many words...
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Toeofdoom wrote:

How the hell do you get that typo???

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Treadmills are for running, not airplanes. That's the trick to this puzzle; airplanes are way bigger than people.

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Right, the plane cannot fly (assuming no wind). Since the wheel movement is matched by the treadmill, no air current is flowing over the wing and no lift is generated.

Related: It is amusing to note that small planes can fly backwards if heading into a strong enough headwind. Since their staying aloft relies only on airspeed, not ground speed, with a stong headwind, they can be pushed backwards relative to the ground yet stay up in 'slow flight'. The things you learn at a school that's mostly pilots...
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Tractor wrote:Right, the plane cannot fly (assuming no wind). Since the wheel movement is matched by the treadmill, no air current is flowing over the wing and no lift is generated.
the air flowing over the wings is produced by the engines. which power the propellors/jets/whatever that are ON THE WINGS. the wheels on a plane are only there to reduce friction (without them the plane would drag along the ground). the wheels are freewheeling and do not do anything to produce lift or thrust.

the plane will take off.

the problem could be re-worded like this: you are in a fighter jet on a carrier. the carrier will match the speed of your wheels, but in the opposite direction. will you take off?
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HYPERiON wrote:
Tractor wrote:Right, the plane cannot fly (assuming no wind). Since the wheel movement is matched by the treadmill, no air current is flowing over the wing and no lift is generated.
the air flowing over the wings is produced by the engines. which power the propellors/jets/whatever that are ON THE WINGS. the wheels on a plane are only there to reduce friction (without them the plane would drag along the ground). the wheels are freewheeling and do not do anything to produce lift or thrust.

the plane will take off.

the problem could be re-worded like this: you are in a fighter jet on a carrier. the carrier will match the speed of your wheels, but in the opposite direction. will you take off?

But the treadmill isn't moving fast enough to counteract what the speed of the wheels would have been if it weren't moving. It is moving at the same speed of the wheels. This is only possible if the plane is stationary, otherwise regardless how fast the treadmill were going, the wheels would be moving faster (or more slowly, if the plane were moving backwards). So the right answer is that either a) this is impossible, as the jets will cause the plane to move forward regardless of the treadmill speed, so the treadmill can't match the speed of the wheels, or b) the plane will not take off, as it is stationary.
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the treadmill will only make the wheels spin twice as fast. nothing more. the only side-effect is friction and maybe exploding tires, but the plane will still take off. after all, a plane can take off in a westerly direction against the earth's rotation, right?
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wut... if the plane is running but isnt actually going anwhere then there will be no air resistance to lift it off the ground... do you ever feel wind in your face while running on a treadmill?

no man, if the plane isnt moving, it wont take off...

but if that treadmill were to stop or suddenly change directions it'd shoot right off, assuming the wheels dont slip...

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Cosmologicon wrote:Alternately, what I first imagined is that when the plane fires its engine as if to go at 100mph, the treadmill spins backward at some speed Nmph, such that the plane and wheels are not moving with respect to the fixed ground, but the wheels are spinning forward as if the plane were moving on fixed ground at speed Nmph. I now think this is ridiculous, because of how huge N must be. It would need to apply enough rolling friction to the wheels in order to slow the whole plane down. In real life, I think it would melt the tires first.

I agree that its ridiculous to suppose that a conveyor belt like that could exist, but that is what we have to suppose because that is what the puzzle says happens.

Of course, if the plane is a harrier then the discussion would be moot.

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Rat wrote:wut... if the plane is running but isnt actually going anwhere then there will be no air resistance to lift it off the ground... do you ever feel wind in your face while running on a treadmill?

no man, if the plane isnt moving, it wont take off...

but if that treadmill were to stop or suddenly change directions it'd shoot right off, assuming the wheels dont slip...
if you were holding a giant fan in each hand you would move quite easily. the wheels on a plane have nothing to do with its propulsion.
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HYPERiON wrote:
Rat wrote:wut... if the plane is running but isnt actually going anwhere then there will be no air resistance to lift it off the ground... do you ever feel wind in your face while running on a treadmill?

no man, if the plane isnt moving, it wont take off...

but if that treadmill were to stop or suddenly change directions it'd shoot right off, assuming the wheels dont slip...
if you were holding a giant fan in each hand you would move quite easily. the wheels on a plane have nothing to do with its propulsion.

Haha. The plane still isn't gonna take off- you don't seem to understand it at all. A plane flies because wind pushes the wings up. There can't be any wind if the wheels aren't going anywhere.

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The way I see it, whatever velocity the plane is going, the surface of the wheels must be moving at the same velocity (not angular velocity of course). So the wheels "may as well be" providing the thrust, no? The treadmill changes velocity (well, speed anyway) when the wheels change speed, and the wheels change speed when the engines' thrust increases or decreases. So the treadmill would be accelerating at the same rate as the acceleration the thrust provides. Therefore, the 'forces' are 'counteracted'. QED

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Just to clarify:

The treadmill matching the (linear) speed of the wheel doesn't mean that the wheel is stationary with respect to an observer beside the treadmill.

For example, if the wheel has circumference 1 and angular velocity 2, then it has linear speed 2 with respect to the treadmill (assuming no slippage). So if the treadmill is moving backwards at speed 1, the wheel has linear speed 1 with respect to an observer.

So the two speeds match, but the aeroplane is still moving forward with respect to an observer.

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Yes. Speed is a scalar, Velocity is a vector.

Let's take speed of the wheel to mean "rotational speed", and convert it to linear by the circumphrence of the wheel.

If the plane is moving at speed X.

And the treadmill surface is moving at speed Y.

Then the "treadmill surface speed" of the plane is |X-Y|.

The "treadmill surface speed" is the speed at which the wheels are turning. If the wheel speed and the surface speed are identical, this means:

Y = |X-Y|

There are two possible solutions:

Y = X-Y
2Y = X
and
Y = Y-X
X = 0

The X=0 case is when the plane is stationary. This is the case that many people assumed we are talking about -- how you use a treadmill when you run on it.

The other possible case is the 2Y=X case. In that case, the treadmill is moving forwards at half of the plane's speed, and the wheels are turning at half of the plane's speed.

In the second case, the plane has no problems taking off.

As such, with the original poster's constrains, it is possible for the plane to take off.

Lastly, the plane could be a VTOL. :)

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I think I'm gonna ask my physics teacher who was in flight school for a definitive answer.
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i think a good way to illustrate this would be as follows:

imagine you're on a plane in mid-air. you lean out the window and start turning the wheels by hand. in fact, you can do it so fast that if you tried to calculate your linear velocity based on their rpm and diameter, you'd get the opposite of the actual velocity of the plane. now add a frictionless treadmill touching the wheels - which makes completely no difference, whatsoever.
so why shouldn't the plane be able to lift off? it will.
but i second the vtol, that would cut this thread short:)
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No, demon. When you're in midair and you spin the wheels, you don't have the friction of the ground, and you're already moving. It's not that the wheels would slow the plane down, they just stop it from accelerating, so they won't let it take off.
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HYPERiON, I feel your pain. Many of the people on this thread would make excellent Verizon customer service reps.

I will try to re-state the explanation from a different angle:

1. Imagine a guy riding a bicycle, wearing a pair of wings.

I sincerely hope that none of you think that this will fly; the only source of propulsion is the wheels/rider, and which simply does not provide enough forward momentum.

2. Imagine, then, a regular car, but with a pair of wings.

Would it fly? Probably not, and if so, not for long.

That is because the only source of propulsion is the wheels pushing against the ground. As soon as the driving wheel loses contact with the ground, there is no more propulsion, and the machine becomes a fucking glider!

3. Now consider how big and heavy an airplane is, how much thrust it takes to get it to lift off, and how long airliners stay up there.

Do you seriously think that all of the energy required for an X-hour flight is provided by the wheels turning at liftoff?!? No! You're fucking loony if you do!

When the Wright brothers made their first flight at Kitty Hawk, it was special because it was the first powered flight, meaning that the aircraft wasn't launched from the ground and glided in the air; it continued to autonomously produce propulsion while in the air.

The only purpose of the wheels (aka "LANDING gear") during takeoff is, as HYPERiON mentioned, to reduce friction while the AIRplane's engines and the air are doing their thing.

4. (Getting a bit back to the issue of the treadmill...) This propulsion is not relative to the ground; it's relative to the air. Usually, the air's position relative to the ground is more or less fixed (not counting wind; but planes take off with the wind coming at them). So in the real world, devoid of infinitely-long treadmills, an airplane moves directly in relation to the air, and indirectly in relation to the ground.

If we were to run a treadmill--"infinitely long" or otherwise--without anything on it, the treadmill would not move the air above it. Perhaps a thin layer, but certainly not a height significant enough to affect the thrust provided by the engines.

Therefore, if a treadmill were to "match the speed of the wheels"--whether this means the wheels don't turn at all, or they would turn twice as fast (which would really mean that the treadmill would have to double its speed again to match their speed, wouldn't it )--it wouldn't matter a slightest bit, because the airplane gets its propulsion from its engines' relationship with the air, not from its wheels' relationship to the ground.

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Look, no one thinks that the wheels are propelling the airplane. Once it's taken off they don't matter at all. But as long as the wheels are going on the treadmill, the plane is being kept in the same place- imagine running on a treadmill- You don't feel the wind against your face. It's the same for a plane on a treadmill- the plane wouldn't be affected by wind- and since the plane relies on air currents to take off, it will stay firmly on the ground.

(if you're getting frustrated, please refer to the first response in the thread)
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This has been discussed.

[quote="Ubertakter"]This post is huge and I knocked it out pretty quickly, so there are likely some mistakes in it. If you find something, post it and I will fix it.

Like others in this thread, I too have broken down and decided to post. First though, there are some misconceptions that have been floating around in this thread that need to be corrected. My writing tends to be blunt, so please keep in mind I am not personally attacking anyone. I am merely trying to explain things. If you read this entire thing and have questions, please ask them. I tried to make my explanation as clear as possible but I may not have been successful. The person who seeks understanding is neither foolish nor stupid.

So, letâ€™s get right to it. First letâ€™s discuss some misconceptions and then discuss the problem. If you just want to see the problem explanation, skip down to the problem section.

Misconceptions

Misconception 1: The engines will somehow suck air over the wing and generate lift.
First, letâ€™s throw out the infinite thrust assumption. I know thatâ€™s how the problem is stated, but this leads us into more problems that we have to ignore that will eventually over-rule the original problem. For instance, you could say that the engine spins at an infinite angular velocity (think revolutions per minute). This means that internal components are moving at the speed of light and that leads us to relativity andâ€¦.bleh. Letâ€™s instead say that the engines generate very large amounts of thrust. More thrust than anything that has ever existed and more thrust than we will ever need.

There is no reason to think that the engine intake is going to pull air over the wing. All of the ducting more or less keeps this from happening. There is also no reason to assume that engine will have any trouble pulling air in due to some low pressure condition since it is exposed to the atmosphere, which is basically infinite in relation to the jet engine. Also, I donâ€™t want to get in to jet engine theory here, but if the turbine did pull air over the wing, there is a good chance it would suck in some of its own exhaust. This creates a condition known as a â€œpop stallâ€
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Oh get out of here, Phesh.

I go through all the trouble of artificially generating controversy to keep this thread alive, and you just copy and paste it into it's grave.

Thanks a lot.
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I did notice you shift sides half way through the thread; I was wondering what was up with that :p
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Owijad wrote:Oh get out of here, Phesh.

I go through all the trouble of artificially generating controversy to keep this thread alive, and you just copy and paste it into it's grave.

Thanks a lot.

What you need is an issue with no scientific backing on either side, which causes controversy.
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Where can we find an issue with no solid scientific evidence one way or the other?

Good Lord, they're almost imposs- wait a minute- Stop the presses!
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i dont believe you.. science is flawed.. the plane wont take off...

the wheels arent going to move at twice the speed of the conveyor if the conveyor is meant to match the speed of the wheels...

in order for the plane to move the wheels would have to move faster than the conveyor...

and i know perfectly well that it's the engines that push the plane along...

plane does not move and does not take off!

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Rat wrote:i dont believe you.. science is flawed.. the plane wont take off...

the wheels arent going to move at twice the speed of the conveyor if the conveyor is meant to match the speed of the wheels...

in order for the plane to move the wheels would have to move faster than the conveyor...

and i know perfectly well that it's the engines that push the plane along...

plane does not move and does not take off!

No.

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